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Can 2 different events in spacetime be connected by both a time-like and a space-like curve?

With flat Minkowski metric I don't think so, because any global orthonormal map $(x,y,z,t)$ of spacetime has its coordinate $t$ strictly increasing for all timelike (continuous) curves.

I'm not so sure with a general curved metric. Are there sufficient conditions on it so that these weird connections cannot happen?

Is it because the light cone of an event $E$ (the collection of all light-like geodesics going through $E$) separates spacetime in 3 connected components : past, future and space?

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    $\begingroup$ This is probably too vague to answer as-is. A few issues are -- do you mean "curve" or "geodesic"? Are you allowing for wormhole topologies, or universes containing exotic matter? Any of your typical "warp drive" or "time machine" spacetimes will have points connected by spacelike AND timelike curves, or they wouldn't have those essential features. $\endgroup$
    – Zo the Relativist
    Commented Jun 13, 2017 at 22:18
  • $\begingroup$ You need some smoothness conditions on the curve to say anything useful. $\endgroup$
    – user107153
    Commented Jun 13, 2017 at 22:30
  • $\begingroup$ Probably not, under some assumptions. One is that the spacetime is connected, smooth and time oriented. It seems to me one can construct a proof from the smoothness, that if you start start from an event in a spacelike tangent direction, to get inside the global region that included both events (assuming the two points can be connected by a purely causal curve) would mean it would have to stop being a spacelike curve somewhere/when. Pure spacelike and causal seems will overlap light cones somewhere along the paths if that's true, and that is not possible. In flat spacetime it is obvious $\endgroup$
    – Bob Bee
    Commented Jun 14, 2017 at 2:10

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In general, it should be true if the dimension of spacetime is at least $3$.

Suppose we have two points $p,q$ in Minkowski space with $p\ll q$, so there is a smooth timelike curve going from $p$ to $q$. I don't have a general formula for the desired spacelike curve, but it should be possible to create by spiraling a spacelike curve in the positive $t$ direction. For simplicity, take $p=(0,0,0,0)$ and $q=(1,0,0,0)$. Clearly $p\ll q$, but there's a spacelike helix winding around the obvious timelike curve connecting the two. Note that a spacelike curve can "go in the timelike direction," just not too strongly. In $\ge 3$ dimensions, you have enough spatial "wiggle room" to get a spacelike curve that grows enough in the timelike direction.

For a general spacetime you could probably take a tubular neighborhood of the timelike curve connecting $p$ and $q$, then doing the same procedure.

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  • $\begingroup$ Indeed, my reasonning above assumed that the spacelike curve was geodesic. How about this stronger question then : 2 events linked by a timelike curve and a spacelike geodesic ? $\endgroup$
    – V. Semeria
    Commented Jun 15, 2017 at 12:21
  • $\begingroup$ @V.Semeria Are you asking if that's possible? It is, assuming you have closed timelike curves (possibly not in general, but I have a specific example). $\endgroup$
    – Ryan Unger
    Commented Jun 15, 2017 at 15:31

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